Finite Group Databasebeta

← #27855

Subgroups — #27855

Order 256.

394 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2855C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2096C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#1927C212yesmaximal, complemented
8#1920C212yesmaximal, complemented
9#1892C212yesmaximal
10#1807C212yesmaximal
11C2×#336C211yesmaximal, characteristic
12C23×C8C2212yes—
13C2×C22⋊Q8C2211yescharacteristic
14C2×#117—21no—
15C22×C42C2211yescharacteristic
16C2×#106C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20#391C2211yescharacteristic
21#391C2211yescharacteristic
22#389C2211yescharacteristic
23#389C2211yescharacteristic
24#387—21no—
25#387—21no—
26#385—21no—
27#384C2211yescharacteristic
28#384C2211yescharacteristic
29#382—21no—
30#381—21no—
31#380—21no—
32#380C2211yescharacteristic
33#380C2212yes—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8—24no—
48C22×C8—24no—
49C22×C8—42no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52C22⋊Q8—24no—
53#117—44no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—24no—
70C2×C42—22no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#106—24no—
76#106—42no—
77#103—24no—
78#103—42no—
79#95—22no—
80#95—22no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88C24C2×D411yescharacteristic
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—21no—
92C2×Q8—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—41no—
107C22×C4SD1614yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4#4114yes—
114C22×C4C2411yescharacteristic
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—48no—
121C2×C8#4114yes—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—44no—
139C22⋊C4—42no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—44no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23#12312yes—
150C23C2×SD1612yes—
151C23#13712yes—
152C23C22×D411yescharacteristic
153Q8—42no—
154Q8—42no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—28no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—44no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4#12312yes—
189C2×C4C2×SD1612yes—
190C2×C4C2×SD1612yes—
191C2×C4#13512yes—
192C2×C4C2×#4112yes—
193C2×C4C2×#4112yes—
194C2×C4H(5,𝔽2)11yescharacteristic
195C2×C42−1+411yescharacteristic
196C8—24no—
197C8—42no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22#48314yes—
205C22C2×#12311yescharacteristic
206C22#53211yescharacteristic
207C22#53711yescharacteristic
208C22C22×SD1611yescharacteristic
209C22C2×#13511yescharacteristic
210C22C2×#13711yescharacteristic
211C22#57511yescharacteristic
212C4—22no—
213C4—22no—
214C4—22no—
215C4—22no—
216C4—41no—
217C4—81no—
218C4—82no—
219C4—81no—
220C4—82no—
221C4—81no—
222C4—81no—
223C4#55112yes—
224C4#55212yes—
225C2—24no—
226C2#249811yescharacteristic
227C2C2×#48311yescharacteristic
228C2#309111yescharacteristic
229C2#316711yescharacteristic
230C2#318111yescharacteristic
231C2#324611yescharacteristic
232C2#324811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #2099C2semidirect
4. #2096C2semidirect
5. #2084C2semidirect
6. #2072C2semidirect
7. #1927C2semidirect
8. #1920C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#28552
#22852
#20992
#20962
#20842
#20722
#19272
#19202
#18922
#18072
C2×#3362